UDC 518:512.25

ON THE CHOICE OF A PARAMETER IN A MINIMIZATION PROBLEM

M. D. Maergoiz

(Presented by Academician L. V. Kantorovich on January 6, 1969)

When applying the method of steepest descent, first proposed in (¹), to the minimization of nonquadratic functionals, it is necessary at each step to determine a parameter by solving a nonlinear equation, i.e., by carrying out an infinite computational procedure. In this connection, in (², ³) special methods were proposed for choosing the parameter that ensure convergence. In the present note a very simple device is proposed for choosing the parameter (for example, on the basis of bisection) for minimizing a strongly convex function in a finite-dimensional space; the convergence of the iterative processes thereby obtained is proved, and a rapidly convergent “hybrid” method is constructed.

Let

\[ X \in E^n;\qquad f(X)\in C^2;\qquad -\infty < f(X^*)=\min_X f(X)<+\infty; \tag{1} \]

the initial approximation \(X^0\) is arbitrary.

Introduce the set \(\mathfrak M_X=\{X: f(X)\leq f(X^0)\}\). Suppose that

\[ m\|u\|^2 < (H(X)u,u)\leq \|H(x)\|\cdot \|u\|^2 < M\|u\|^2; \tag{2} \]

for all \(X\in\mathfrak M_X,\ u\in E^n\), where

\[ H(X)=\left(\frac{\partial^2 f(X)}{\partial x_i\partial x_j}\right)^n_{i,j=1}. \]

We describe a step of any of the iterative processes:

\[ \left\{ \begin{aligned} &t_0^k=t>0;\quad X^{-t_j^k}=X^k-t_j^k l^k;\\ &\text{if } f(X^{-t_j^k})\geq f(X^k),\text{ then } t_{j+1}^k=t_j^k/p; \end{aligned} \right. \tag{3} \]

otherwise:

\[ t^k=t_j^k;\qquad \tau^k=t^k/p;\qquad p>1;\qquad X^{k+1}=X^k-\tau^k l^k, \tag{4} \]

where \(l^k\) satisfies the relation

\[ \gamma_1\|\nabla f(X^k)\|^2 < (\nabla f(X^k),l^k)\leq \|\nabla f(X^k)\|\cdot \|l^k\| < \gamma_2\|\nabla f(X^k)\|^2; \tag{5} \]

\[ 0<\gamma_1;\qquad k=0,1,2,\ldots \]

For \(p=2\) we obtain the bisection device.

Theorem 1. If conditions (1), (2) are fulfilled, then any of the iterative processes (3), (4), (5) converges, i.e.

\[ \lim_{k\to\infty}\|X^k-X^*\|=0. \tag{6} \]

Proof. For any \(X^k\) \((\|\nabla f(X^k)\|>0)\), consider

\[ \varphi_k(t)=(\nabla f(X^k-tl^k),l^k);\qquad \varphi_k(0)=(\nabla f(X^k),l^k)>\gamma_1\|\nabla f(X^k)\|^2>0. \tag{7} \]

According to (1), (2),

\[ \varphi'_k(t)=-(H(X^k-tl^k)l^k,l^k)<0, \tag{8} \]

the function \(\varphi_k(t)\) is monotonically decreasing and has one positive root \(a^k\). From (3), (4),

\[ f(X^k-t^k l^k)-f(X^k) = -\int_0^{t^k}(\nabla f(X^k-tl^k),l^k)\,dt = -\int_0^{t^k}\varphi_k(t)\,dt<0, \tag{9} \]

\[ \int_0^{t^k}\varphi_k(t)\,dt>0. \]

Taking into account (7), (8),

\[ \int_0^{\tau^k}\varphi_k(t)\,dt = \int_0^{t^k/p}\varphi_k(t)\,dt>0; \]

\[ f(X^{k+1})-f(X^k) = f(X^k-\tau^k l^k)-f(X^k) = -\int_0^{\tau^k}\varphi_k(t)<0; \qquad f(X^{k+1})<f(X^k). \tag{10} \]

Thus, the sequence \(\{f(X^k)\}\) is monotonically decreasing and bounded below \((f(X^k)\geq f(X^*))\).

Suppose that the sequence \(\{f(X^k)\}\) does not converge to the minimum value of the function, i.e.,

\[ \lim_{k\to\infty} f(X^k)=\bar f>\min_X f(X)=f(X^*). \tag{11} \]

In this case there is a \(\Delta>0\) such that for all \(k\) it will be true that

\[ \|\nabla f(X^k)\|>\Delta. \tag{12} \]

In order to obtain a contradiction to assumption (11), we first estimate the parameter \(\tau^k\).

Taking into account (3), (4), \(f(X^k-t^k l^k)<f(X^k)\); \(f(X^k-pt^k l^k)\geq f(X^k)\). By continuity there is a number

\[ \lambda^k=(1+\theta^k(p-1))t^k;\qquad 0<\theta^k\leq 1, \tag{13} \]

such that

\[ f(X^k-\lambda^k l^k)=f(X^k);\qquad X^k-\lambda^k l^k\in \mathfrak M_X; \]

\[ f(X^k-\lambda^k l^k)-f(X^k) = -\lambda^k(\nabla f(X^k),l^k) + \frac{(\lambda^k)^2}{2}(H(y^k)l^k,l^k)=0; \tag{14} \]

\[ \lambda^k= \frac{2(\nabla f(X^k),l^k)}{(H(y^k)l^k,l^k)}; \qquad Y^k=X^k-\omega^k\lambda^k l^k\in\mathfrak M_X; \qquad 0<\omega^k<1. \]

In connection with (2), (3), (4), (5), (13), (14),

\[ \frac{2\gamma_1}{\gamma_2^2 M}<\lambda^k<\frac{2\gamma_2}{\gamma_1^2 m}; \qquad \min\left(t,\frac{2\gamma_1}{p\gamma_2^2 M}\right)<t^k<\frac{2\gamma_2}{\gamma_1^2 m}; \]

\[ \min\left(\frac{t}{p},\frac{2\gamma_1}{p^2\gamma_2^2 M}\right)<\tau^k<\frac{2\gamma_2}{p\gamma_1^2 m}. \tag{15} \]

Let us estimate from below

\[ \int_0^{\tau^k}(\nabla f(X^k-tl^k),l^k)\,dt = \int_0^{\tau^k}\varphi_k(t)\,dt. \]

On the basis of (1), (2), (4), (5), (13), (14),

\[ -M\gamma_2^2\|\nabla f(X^k)\|^2 < \dot\varphi_k(t) = -(H(X^k-tl^k)l^k,l^k) < -m\gamma_1^2\|\nabla f(X^k)\|^2; \]

\[ 0\leq t\leq\lambda^k;\qquad X^k-tl^k\in\mathfrak M_X. \tag{16} \]

\[ -M\gamma_2^2\|\nabla f(X^k)\|^2 t+\varphi_k(0) \leq \varphi_k(t) \leq -m\gamma_1^2\|\nabla f(X^k)\|^2 t+\varphi_k(0); \qquad 0\leq t\leq\tau^k; \]

\[ -M\gamma_2^2\|\nabla f(X^k)\|^2(t-\tau^k)+\varphi_k(\tau^k) \leq \varphi_k(t) \leq \]

\[ \leq -m\gamma_1^2\|\nabla f(X^k)\|^2(t-\tau^k)+\varphi_k(\tau^k); \qquad \tau^k\leq t\leq\lambda^k. \tag{17} \]

Suppose first that \(\tau^k \leqslant a^k\). According to (7), (8), (14), (16), (12)

\[ \int_0^{\tau^k}\varphi_k(t)\,dt > \int_0^{\min(t/p,\,2\gamma_1/p^2\gamma_2^2M)} \left(-M\gamma_2^2\|\nabla f(X^k)\|^2t+\varphi_k(0)\right)\,dt > \]

\[ > \int_0^{\min(t/p,\,2\gamma_1/p^2\gamma_2^2M)} \left(-M\gamma_2^2\|\nabla f(X^k)\|^2t+ \min(t\gamma_2^2M/2\gamma_1,\,\frac{1}{p})\varphi_k(0)\right)\,dt > \]

\[ > \min\left(\frac{2\gamma_1^2}{p^4\gamma_2^2M},\, \frac{1}{2}\frac{t^2\gamma_2^2M}{p^2}\right)\Delta^2=S_1. \tag{18} \]

Now suppose that \(\tau^k \geqslant a^k\). According to (7), (8), (15), 17) \(\varphi_k(\tau^k)\leqslant 0\);

\[ \int_0^{\tau^k}\varphi_k(t)\,dt = \int_0^{\lambda^k}\varphi_k(t)\,dt - \int_{\tau^k}^{\lambda^k}\varphi_k(t)\,dt = f(X^k)-f(X^k-\lambda^kl^k)- \]

\[ -\int_{\tau^k}^{\lambda^k}\varphi_k(t)\,dt = -\int_{\tau^k}^{\lambda^k}\varphi_k(t)\,dt \geqslant \int_{\tau^k}^{\lambda^k} \left(m\gamma_1^2\|\nabla f(X^k)\|^2(t-\tau^k)-\varphi_k(\tau^k)\right)\,dt \geqslant \]

\[ \geqslant \int_{\tau^k}^{\lambda^k} m\gamma_1^2\|\nabla f(X^k)\|^2(t-\tau^k)\,dt = \frac{m\gamma_1^2\|\nabla f(X^k)\|^2}{2}(\lambda^k-\tau^k)^2 > \]

\[ > \frac{m\gamma_1^2}{2}(t^k-\tau^k)^2\Delta^2 = \frac{m\gamma_1^2(p-1)^2}{2} \left(\min\left(\frac{t}{p},\,\frac{2\gamma_1}{p^3\gamma_2^2M}\right)\right)^2 \Delta^2 = S_2. \tag{19} \]

Finally,

\[ \int_0^{\tau^k}\varphi_k(t)\,dt > \min(s_1,s_2)=s>0; \]

\[ f(X^{k+1})=f(X^k)-\int_0^{\tau^k}\varphi_k(t)\,dt < f(X^k)-s < f(X^0)-(k+1)s; \tag{20} \]

\[ \lim_{k\to\infty} f(X^k)=-\infty < f(X^*)=\min_X f(X), \]

which contradicts (11).

Thus, for any \(\varepsilon>0\) there exists such a \(K(\varepsilon)\) that for all \(k\geqslant K(\varepsilon)\) the inequality \(\|\nabla f(X^k)\|<\varepsilon\) will hold. Hence

\[ \nabla f(X^k)=\nabla f(X^k)-\nabla f(X^*)=H(Z^k)(X^k-X^*); \]

\[ Z^k=X^*+\theta^k(X^k-X^0)\in\mathfrak{M}_X,\quad 0<\theta^k<1; \tag{21} \]

\[ X^k-X^*=(H(Z^k))^{-1}\nabla f(X^k);\qquad \|X^k-X^*\|<\varepsilon/m, \]

which proves the theorem.

Choose \(l^k=(H(X^k))^{-1}\nabla f(X^k)\).

\[ \frac{\|\nabla f(X^k)\|^2}{M} < (\nabla f(X^k),l^k) = (\nabla f(X^k),(H(X^k))^{-1},\nabla f(X^k))\leqslant \tag{22} \]

\[ \leqslant \|(H(X^k))^{-1}\|\cdot\|\nabla f(X^k)\| < \frac{\|\nabla f(X^k)\|^2}{m}; \]

\[ \min\left(t,\frac{2m^2}{pM^2}\right)<t^k<\frac{2M^2}{m^2}; \qquad \frac{2m^2}{pM^2}<2;\qquad \frac{2M^2}{m^2}>2; \tag{23} \]

\[ \min\left(t/p,\frac{2m^2}{p^2M^2}\right)<\tau^k<\frac{2M^2}{pm^2}. \]

Theorem 2. If \(0<t<2\), then there exists a neighborhood
\[ W_\delta(X^\circ)=\{X:\|X-X^\circ\|\leq \delta\}\subset \mathfrak M_X \]
such that for
\[ X^k\in W_\delta(X^\circ);\quad X^k_{(t)}=X^k-t(H(X^k))^{-1}\nabla f(X^k) \tag{24} \]
the inequality \(f(X^k_{(t)})<f(X^k)\) holds.

Proof. From (24) we have
\[ X^k_{(t)}-X^k=-t(H(X^k))^{-1}(\nabla f(X^k)-\nabla f(X^\circ))= \]
\[ =-t(H(X^k))^{-1}H(\bar X^k)(X^k-X^\circ); \]
\[ \bar X^k=X^\circ+\bar\theta^k(X^k-X^\circ)\in W_\delta(X^\circ)\subset \mathfrak M_X; \tag{25} \]
\[ 0<\bar\theta^k<1;\quad \|X^k_{(t)}-X^k\|\leq \frac{2M}{m}\,\delta=\delta'. \]

Let us choose a number
\[ 0<\varepsilon<\frac{(2-t)m^2}{Mt}. \tag{26} \]
In view of (1), for \(\varepsilon\) there exists such a \(\delta'>0\), and hence also \(\delta\), that whenever the inequality
\[ \|X-X^k\|\leq \frac{2M}{m}\,\delta=\delta' \]
is satisfied, we have
\[ \|H(X)-H(\bar X^k)\|<\varepsilon; \]
\[ f(X^k_{(t)})=f(X^k)-t(\nabla f(X^k),(H(X^k))^{-1}\nabla f(X^k))+ \frac{t^2}{2}(H(\bar X^k_{(t)})\times \]
\[ \times (H(X^k))^{-1}\nabla f(X^k),(H(X^k))^{-1}\nabla f(X^k)); \quad \bar X^k_{(t)}=X^k+\bar\theta^k_{(t)}(X^k-X^k_{(t)}); \]
\[ 0<\bar\theta^k_{(t)}<1;\quad \|\bar X^k_{(t)}-X^k\|\leq \frac{2M}{m}\,\delta=\delta'; \]
\[ f(X^k_{(t)})=f(X^k)-\frac{t}{2}(\nabla f(X^k),(H(X^k))^{-1}\nabla f(X^k))(2-t)+ \]
\[ +\frac{t^2}{2}((H(\bar X^k_{(t)})-H(X^k))(H(X^k))^{-1}\nabla f(X^k),(H(X^k))^{-1}\nabla f(X^k))< \]
\[ <f(X^k)-\frac{t}{2}\frac{\|\nabla f(X^k)\|^2}{M}(2-t) +\frac{t^2}{2}\varepsilon\frac{\|\nabla f(X^k)\|^2}{m^2}= \]
\[ =f(X^k)-\frac{t^2\|\nabla f(X^k)\|^2}{2m^2} \left(\frac{(2-t)m^2}{Mt}-\varepsilon\right)<f(X^k), \]
which completes the proof.

In (3) choose \(t=p,\ 1<p<2\). In this case the iterative process
\[ X^{k+1}=X^k-\tau^k(H(X^k))^{-1}\nabla f(X^k) \tag{27} \]
converges on the basis of (22) and Theorem 1, and according to Theorem 2 there exists such a \(K\) that for all \(k\geq K\) the equality
\[ \tau^k=1 \tag{28} \]
holds.

Theorem 3. The iterative process (27) converges with superlinear rate.

Proof. From (27), (28), for \(k\geq K\) we have
\[ X^{k+1}-X^\circ=X^k-X^\circ-(H(X^k))^{-1}(\nabla f(X^k)-\nabla f(X^\circ))= \]
\[ =X^k-X^\circ-(H(X^k))^{-1}H(\bar X^k)(X^k-X^\circ)=(H(X^k))^{-1}(H(X^k)- \tag{29} \]
\[ -H(\bar X^k))(X^k-X^\circ); \quad \|X^{k+1}-X^\circ\|\leq \frac{\varepsilon}{m}\|X^k-X^\circ\|, \]
i.e. the assertion of the theorem.

If \(f(X)\in C^3\), then the iterative process (27) with the choice of the parameter according to (3), (4) \((1<t=p<2)\) converges, as does Newton’s method for solving the nonlinear system \(\nabla f(X)=0\), with quadratic rate (see \((^4)\)).

Institute of Cybernetics
Academy of Sciences of the Ukrainian SSR

Received
30 XII 1968

CITED LITERATURE

1. L. V. Kantorovich, DAN, 48, No. 7, 483 (1945).
2. I. M. Glazman, DAN, 154, No. 5, 1011 (1964).
3. A. A. Goldstein, SIAM, J. on Control, A, 3, No. 1, 147 (1965).
4. L. V. Kantorovich, Vestn. Leningrad Univ., 2, No. 7, 68 (1957).